The Discontinuity‐Enriched Finite Element Method

Numerical Meth Engineering

Sun

et al. 2020

Smoothed finite element method with the node-based strain smoothing domains (NS-FEM) is remarkable for the upper-bound feature and insensitivity to the volumetric locking. As a mesh-based methodology, its application is limited by the burdensome meshing for which elements align with the physical domain. We extend the strain smoothing in NS-FEM to the numerical manifold method (NMM) with unfitted meshes, and propose a novel methodology named physical patch-based smoothed NMM. Benchmark problems demonstrate the optimal convergence, stability in term of the condition number, upper-bound property, suppression of the volumetric locking, as well as the stress stability.

Section: Introductionmentioning

confidence: 99%

Smoothed numerical manifold method with physical patch‐based smoothing domains for linear elasticity

Liu

Numerical Meth Engineering

Sun

et al. 2020

“…Furthermore, enrichment functions are exactly zero at original mesh nodes. Therefore, original mesh nodes retain their physical meaning and essential boundary conditions can be enforced directly on non-matching edges (Cuba-Ramos et al 2015;Aragón and Simone 2017;van den Boom et al 2019a). It was shown that IGFEM is optimally convergent under mesh refinement for problems without singularities (Soghrati et al 2012a, b).…”

Section: Introductionmentioning

confidence: 99%

“…IGFEM was later developed into the Hierarchical Interface-enriched Finite Element Method (HIFEM) (Soghrati 2014) that allows for intersecting discontinuities, and into the Discontinuity-Enriched Finite Element Method (DE-FEM) (Aragón and Simone 2017), which provides a unified formulation for both strong and weak discontinuities (i.e., discontinuities in the field and its gradient, respectively). DE-FEM, which inherits the same advantages of IGFEM over X/GFEM, has successfully been applied to problems in fracture mechanics (Aragón and Simone 2017;Zhang et al 2019a) and fictitious domain or immersed boundary problems with strongly enforced essential boundary conditions (van den Boom et al 2019a). A drawback of IGFEM is that quadratic enrichment functions are needed when the method is applied to background meshes composed of bilinear quadrangular elements (Aragón et al 2020).…”

Section: Introductionmentioning

confidence: 99%

An interface-enriched generalized finite element method for level set-based topology optimization

Boom

Keulen

et al. 2020

Struct Multidisc Optim

Self Cite

During design optimization, a smooth description of the geometry is important, especially for problems that are sensitive to the way interfaces are resolved, e.g., wave propagation or fluid-structure interaction. A level set description of the boundary, when combined with an enriched finite element formulation, offers a smoother description of the design than traditional density-based methods. However, existing enriched methods have drawbacks, including ill-conditioning and difficulties in prescribing essential boundary conditions. In this work, we introduce a new enriched topology optimization methodology that overcomes the aforementioned drawbacks; boundaries are resolved accurately by means of the Interface-enriched Generalized Finite Element Method (IGFEM), coupled to a level set function constructed by radial basis functions. The enriched method used in this new approach to topology optimization has the same level of accuracy in the analysis as the standard finite element method with matching meshes, but without the need for remeshing. We derive the analytical sensitivities and we discuss the behavior of the optimization process in detail. We establish that IGFEM-based level set topology optimization generates correct topologies for well-known compliance minimization problems.

“…For the next hierarchy level, only the coordinate x 4 is located on Γ i , but here, both weak ( α ) and strong ( β ) DOFs are present. Because β physically represents the crack opening displacement, their values are readily available once the jump in the displacement is known, ie,

{bold-italicβ}_{21} = ⟦ bold-italicu false({bold-italicx}_{4} false) ⟧ = \overset{bold-italicu}{true} false({bold-italicx}_{4} false) {false|}_{{normalΓ}_{normalc}^{+}} - \overset{bold-italicu}{true} false({bold-italicx}_{4} false) {false|}_{{normalΓ}_{normalc}^{-}}

. Then, solving for α at x 4 follows the same procedure just described for α 11 and α 12 .…”

Section: Formulationmentioning

confidence: 99%

“…Cuba‐Ramos et al demonstrated IGFEM as an immersed boundary method, by using Lagrange multipliers to weakly impose Dirichlet boundary conditions on non‐matching boundaries. More recently, Aragón and Simone introduced the Discontinuity‐Enriched FEM (DE‐FEM) as a generalization of IGFEM to treat both weak and strong discontinuities with a unified formulation. In addition to the flexibility of dealing with both discontinuity types, DE‐FEM inherits all of the virtues of IGFEM/HIFEM: The construction of both weak and strong enrichment functions is straightforward, as they are based on the standard Lagrange shape functions of integration elements; Because enrichment functions vanish at original mesh nodes, the Kronecker‐delta property is maintained in standard nodes, allowing essential boundary conditions to be applied in the same way as in standard FEM; With the use of a diagonal preconditioner, or a proper scaling factor for the enrichment functions, the formulation used for treating weak discontinuities is stable, ie, the condition number increases at the same rate as that of standard FEM under mesh refinement; A hierarchical implementation of DE‐FEM can analyze multiple discontinuities and n ‐junctions within a single element.…”

Section: Introductionmentioning

confidence: 99%

A stable interface‐enriched formulation for immersed domains with strong enforcement of essential boundary conditions

Boom

Numerical Meth Engineering

Keulen

et al. 2019

Self Cite

Aims and ScopeThe International Journal for Numerical Methods in Engineering publishes original papers describing significant, novel developments in numerical methods that are applicable to engineering problems. The Journal is known for welcoming contributions in a wide range of areas in computational engineering, including computational issues in model reduction, uncertainty quantification, verification and validation, inverse analysis and stochastic methods, optimisation, element technology, solution techniques and parallel computing, damage and fracture, mechanics at micro and nano-scales, low-speed fluid dynamics, fluid-structure interaction, electromagnetics, coupled diffusion phenomena, and error estimation and mesh generation. It is emphasized that this is by no means an exhaustive list, and particularly papers on multi-scale, multi-physics or multi-disciplinary problems, and on new, emerging topics are welcome. Manuscripts should have sufficient original numerical content, and generate new knowledge that is applicable to general classes of engineering problems, and not be limited to applications of existing methods, or propose incremental improvements to existing methods. The journal publishes full-length papers, which should normally be less than 25 journal pages in length, and short communications, which can be at most 8 journal pages. Discussions of papers in print can be published, but two-part papers will not be considered for review.